These notes are drawn from lectures given at University College Cork in the spring of 2006, for a �rst year introduction to linear algebra. The course aims for a complete proof of the spectral theorem. Problems appear throughout the text, which you must learn to solve. They often provide vital results used in the course. Most of these problems have hints, particularly the more important ones. There are also review problems at the end of each section, and you should try to solve a few from each section. Try to solve each problem �rst before looking up the hint.

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